H\"older Continuity of Harmonic Quasiconformal Mappings

TL;DR

This paper proves that harmonic quasiconformal mappings with boundary H"older continuity are also H"older continuous inside, under certain domain conditions, advancing understanding of boundary behavior in harmonic quasiconformal maps.

Contribution

It establishes boundary-to-interior H"older continuity for harmonic quasiconformal mappings on uniformly perfect domains, allowing some boundary thinness.

Findings

H"older continuity on the boundary implies interior H"older continuity.

Results apply to uniformly perfect bounded domains with some boundary thinness.

Open problem remains for general bounded domains.

Abstract

We prove that for harmonic quasiconformal mappings $\alpha$-H\"older continuity on the boundary implies $\alpha$-H\"older continuity of the map itself. Our result holds for the class of uniformly perfect bounded domains, in fact we can allow that a portion of the boundary is thin in the sense of capacity. The problem for general bounded domains remains open.